| Brendan McKay writes: (Start)
"It would be possible to find the counts for n=9 and n=10 using the method of my paper in JCD: http://cs.anu.edu.au/~bdm/papers/ls_final.pdf. For n=10 it is probably a 24-digit number. I'll explain the method I used. See the paper above for terminology.
"Is(L) is the autotopism group. Also define the group RC(L) of all autotopisms for which the symbols component is the identity. For any Latin square L we have:
"The isotopy class containing L contains (n!)^3/|Is(L)| squares.
"The RC-equivalence class containing L contains (n!)^2/|RC(L)| squares.
"If L and L' are isotopic then |RC(L)| = |RC(L')|. Therefore the number of RC-equivalence classes in the isotopy class of L is n!*|RC(L)|/|Is(L)|. I modified an existing program slightly to find |RC(L)|/|Is(L)|. and applied it to one square from each isotopy class. The sum of n!*|RC(L)|/|Is(L)| is the total number of RC-equivalence classes. " (End)
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